Piezoelectric driving optimal trajectory tracking control method based on adaptive step length

Abstract

The invention discloses a piezoelectric driving optimal trajectory tracking control method based on adaptive step length, which comprises the following steps of: performing mathematical modeling on a piezoelectric ceramic driver by adopting an electromechanical coupling nonlinear power system model to obtain a piezoelectric ceramic driver model; converting the piezoelectric ceramic driver model into a linear controllable standard subsystem; taking the expected displacement x * (0) when the time t is equal to 0 as the initial actual tracking displacement x (0); feedback control u is solved through a linear controllable specification subsystem; the initial actual tracking displacement x (0) is substituted into the linear controllable specification subsystem, and the actual tracking displacement is solved through a Runge-Kutta-Filbert method with the self-adaptive step length; and determining a performance index function of the linear controllable specification subsystem, substituting the feedback control u and the actual tracking displacement into the performance index function, and solving an optimal performance index of the piezoelectric ceramic driver model. According to the method, the calculation efficiency of the piezoelectric ceramic driver can be improved while high precision is kept in the calculation process.

Description

Technical Field

[0001] The invention relates to the technical field of piezoelectric ceramic driver control, and in particular to a piezoelectric driver optimal trajectory tracking control method based on adaptive step length. Background Art

[0002] Piezoelectric ceramic actuators are widely used in ultra-precision machining, biomedical engineering, nanotechnology and other fields due to their high stiffness, high frequency response, small size and light weight. However, the nonlinear characteristics of piezoelectric ceramic actuators, especially hysteresis and creep phenomena, limit their application in high-precision displacement control. These nonlinear characteristics not only affect the control accuracy of the system, but may also cause system instability, becoming a key challenge for piezoelectric actuators to further improve their performance. In the past control methods, PID control was widely used, but its performance was difficult to meet high-precision requirements in the face of complex nonlinear hysteresis characteristics. In order to accurately describe the nonlinear hysteresis characteristics of piezoelectric ceramic actuators, researchers have proposed a variety of models, such as the Preisach model, the Wiener model and the hysteresis creep model based on the Prandtl-Ishlinskii operator. Although these models can better describe the hysteresis behavior of piezoelectric ceramic actuators, most of them can only deal with static hysteresis characteristics and require a large amount of experimental data to support them.

[0003] The CNLP (electromechanical coupling nonlinear lumped parameter) model is a model that integrates the nonlinear hysteresis characteristics of the piezoelectric stack and the dynamics of the driver proposed by Bin Yang et al. in the Journal of Automatica Sinica. This model converts it into a linear controllable subsystem through the input-output linearization principle of differential geometry. Combined with the linear quadratic form (LQ) method, it can effectively improve the displacement tracking accuracy of the driver and provide a new idea for the nonlinear dynamic modeling of piezoelectric ceramic drivers. However, these control algorithms still have the problems of complex calculations and limited control accuracy in practical applications. In order to solve the above problems, Mai Heng et al. used the traditional fourth-order Runge-Kutta method (hereinafter referred to as "RK4") to calculate the tracking displacement in the system in the patent with the authorization announcement number CN108777553B and the patent name of a piezoelectric ceramic driver control method based on the Runge-Kutta method. However, in a short period of time, due to the fixed step size used in the traditional RK4 method, the accuracy of the piezoelectric ceramic driver with complex calculations still needs to be improved.

[0004] Therefore, in the field of micro-nano precision control, an algorithm that can improve the displacement tracking accuracy of piezoelectric actuators is still needed to adapt to complex ECNLP models. Summary of the invention

[0005] The purpose of the present invention is to overcome the shortcomings of the prior art and provide a piezoelectric drive optimal trajectory tracking control method based on adaptive step size. This method improves the computational efficiency and control accuracy of the piezoelectric ceramic driver by adaptively adjusting the step size while maintaining high accuracy during the calculation process. Compared with the RK4 algorithm proposed by Mai Heng et al. with a fixed step size, the Runge Kutta-Feldberg method can dynamically adjust the step size according to the system state, so as to better cope with the complexity of nonlinear dynamic systems, especially in the field of micro-nano precision control. It shows superior performance. This method significantly improves the displacement tracking capability of the piezoelectric ceramic driver and provides a new solution for the application of high-precision positioning technology.

[0006] To achieve the above purpose, the technical solution provided by the present invention is:

[0007] A piezoelectric drive optimal trajectory tracking control method based on adaptive step size, comprising:

[0008] The piezoelectric ceramic actuator is mathematically modeled using an electromechanical coupling nonlinear dynamic system model to obtain a piezoelectric ceramic actuator model;

[0009] Convert the piezoelectric ceramic actuator model into a linear controllable canonical subsystem;

[0010] The expected displacement x at time t = 0 * (0) as the initial actual tracking displacement x(0);

[0011] The feedback control u is obtained through the linear controllable canonical subsystem;

[0012] Substitute the initial state value x(0) into the linear controllable gauge subsystem and use the Runge-Kutta-Feldberg method with adaptive step size to obtain the actual tracking displacement.

[0013] The performance index function of the linear controllable specification subsystem is determined, and the feedback control u and the actual tracking displacement are substituted into the performance index function to obtain the optimal performance index of the piezoelectric ceramic actuator model.

[0014] Furthermore, after mathematically modeling the piezoelectric ceramic actuator using the electromechanical coupling nonlinear dynamic system model, the obtained piezoelectric ceramic actuator model is as follows:

[0015]

[0016] in, is the derivative of X, is the state variable, U in is the control variable, F is the state variable X and the control variable U of the piezoelectric ceramic actuator model in The relevant function matrix, G is the weight matrix, x is the actual tracking displacement, is the derivative of x.

[0017] Furthermore, the transformed linear controllable canonical subsystem is as follows:

[0018]

[0019] in, u is feedback control.

[0020] Furthermore, the formula for calculating the feedback control u is as follows:

[0021] u=-R -1 B T P LQ Z+R -1 B T G g (3)

[0022] R is the weight matrix, constant matrix P LQ It is the only symmetric positive definite solution to the Riccati algebraic equation;

[0023] A T P LQ +P LQ AP LQ BR -1 B T P LQ +C T QC=0 (4)

[0024] C is the piezoelectric ceramic driver parameter, and its value is 1.2×10 -6 F, matrix G g for:

[0025] G g =[P LQ BR -1 B T -A T ] -1 C T Qx * (5)

[0026] x * is the expected displacement.

[0027] Furthermore, the actual tracking displacement is obtained by using the Runge-Kutta-Feldberg method with adaptive step size, including:

[0028] A1. Set initialization parameters: the number of allowed iterations temp, the minimum scaling factor min_scale, the maximum scaling factor max_scale, the error tolerance ε, the system initial state value x0 = x(0), the independent variable t = t0, i = 1;

[0029] A2. Obtain the current state value and the generated error;

[0030] A3, determine whether the error generated in the previous step A2 is equal to 0, if so, directly go to step A5, otherwise go to step A4;

[0031] A4: Check whether i is equal to temp. If so, go to step A5. Otherwise,

[0032] Update the scaling factor s, step size h, time t, and i:

[0033] in,

[0034] If min_scale<s<max_scale, then err i is the current error, h = h·s, t = t+h, i = i+1;

[0035] After updating, return to step A2;

[0036] A5. Use the latest state value as the actual tracking displacement x(t).

[0037] Furthermore, the calculation process of the state value corresponding to the i-th iteration and the generated error includes:

[0038] k1=hf(t i ,x i )

[0039]

[0040] Where h is the step size.

[0041] Furthermore, the performance index function of the determined linear controllable canonical subsystem is as follows:

[0042]

[0043] Among them, x * is the desired displacement, R and Q are weight matrices, u is the feedback control, and x is the actual tracking displacement.

[0044] Furthermore, the optimal performance indicators of the piezoelectric ceramic driver model are obtained, including:

[0045] Transform the performance indicator function into:

[0046]

[0047] Let M tend to positive infinity, according to

[0048]

[0049] Right now

[0050] When M approaches positive infinity, substitute the feedback control u and the actual tracking displacement into formula (8) to obtain the limit That is, the optimal performance index of the piezoelectric ceramic driver model is obtained.

[0051] Compared with the prior art, the principles and advantages of this technical solution are as follows:

[0052] 1. By adaptively adjusting the step size based on the Runge-Kutta-Feuerberg method (RKF45), the computational efficiency and control accuracy of the piezoelectric ceramic actuator are improved while maintaining high accuracy during the calculation process.

[0053] 2. The error tolerance ε is adjusted according to the accuracy requirements, so that the calculation accuracy is better than the fourth-order Runge-Kutta method and the calculation speed is better than the fifth-order Runge-Kutta method, thereby further improving the calculation accuracy and speed of the piezoelectric actuator displacement tracking, so that the system obtains better control effect. BRIEF DESCRIPTION OF THE DRAWINGS

[0054] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the services required for use in the embodiments or the prior art descriptions are briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0055] Figure 1 It is a principle flow chart of a piezoelectric drive optimal trajectory tracking control method based on adaptive step size of the present invention;

[0056] Figure 2 A schematic diagram of solving the actual tracking displacement in a piezoelectric drive optimal trajectory tracking control method based on adaptive step size according to the present invention;

[0057] Figure 3 Flow chart of solving the actual tracking displacement using the Runge-Kutta-Feldberg method with adaptive step size;

[0058] Figure 4 It is a schematic diagram comparing the error between RKF45 and ode45 and the error between RK4 and ode45;

[0059] Figure 5 Schematic diagram for comparison between ECNLP models RK4 and RKF45 in displacement tracking control system. DETAILED DESCRIPTION

[0060] The present invention will be further described below in conjunction with specific embodiments:

[0061] like Figure 1 to Figure 2 As shown, the piezoelectric drive optimal trajectory tracking control method based on adaptive step size described in this embodiment includes the following steps:

[0062] S1. The piezoelectric ceramic driver is mathematically modeled using the electromechanical coupling nonlinear dynamic system model, and the following piezoelectric ceramic driver model is obtained:

[0063]

[0064] in, is the derivative of X, is the state variable, U in is the control variable, F is the state variable X and the control variable U of the piezoelectric ceramic actuator model in The relevant function matrix, G is the weight matrix, x is the actual tracking displacement, is the derivative of x.

[0065] S2. Convert the piezoelectric ceramic actuator model into a linear controllable canonical subsystem:

[0066]

[0067] in, u is feedback control.

[0068] S3. The expected displacement x at time t = 0 * (0) as the initial actual tracking displacement x(0);

[0069] S4. The feedback control u is obtained through the linear controllable specification subsystem, and the formula is as follows:

[0070] u=-R -1 B T P LQ Z+R -1 B T G g (3)

[0071] R is the weight matrix, constant matrix P LQ It is the only symmetric positive definite solution to the Riccati algebraic equation;

[0072] A T P LQ +P LQ AP LQ BR -1 B T P LQ +C T QC=0 (4)

[0073] C is the piezoelectric ceramic driver parameter, and its value is 1.2×10-6 F, matrix G g for:

[0074] G g =[P LQ BR -1 B T -A T ] -1 C T Qx * (5)

[0075] x * is the expected displacement.

[0076] S5. Substitute the initial state value x(0) into the linear controllable gauge subsystem and use the Runge-Kutta-Feldberg method with adaptive step size to calculate the actual tracking displacement, such as Figure 3 As shown, the specific process includes:

[0077] A1. Set initialization parameters: the number of allowed iterations temp, the minimum scaling factor min_scale, the maximum scaling factor max_scale, the error tolerance ε, the step size h, the system initial state value x0 = x(0), the independent variable t = t0, i = 1;

[0078] A2. Obtain the current state value and the generated error;

[0079] The calculation process of the state value and the error corresponding to the i-th iteration includes:

[0080] k1=hf(t i ,x i )

[0081]

[0082] A3, determine whether the error generated in the previous step A2 is equal to 0, if so, directly go to step A5, otherwise go to step A4;

[0083] A4: Check whether i is equal to temp. If so, go to step A5. Otherwise,

[0084] Update the scaling factor s, step size h, time t, and i:

[0085] in,

[0086] If min_scale<s<max_scale, then (here h is before update), err i is the current error, h = h·s, t = t+h (here h is the updated one), i = i+1;

[0087] After updating, return to step A2;

[0088] A5. Use the latest state value as the actual tracking displacement x(t).

[0089] S6. Determine the performance index function of the linear controllable specification subsystem, and substitute the feedback control u and the actual tracking displacement into the performance index function, so as to find the optimal performance index of the piezoelectric ceramic driver model.

[0090] In this step, the performance index function of the determined linear controllable specification subsystem is as follows:

[0091]

[0092] Among them, x * is the desired displacement, R and Q are weight matrices, u is the feedback control, and x is the actual tracking displacement.

[0093] Find the optimal performance indicators of the piezoelectric ceramic actuator model, including:

[0094] Transform the performance indicator function into:

[0095]

[0096] Let M tend to positive infinity, according to

[0097]

[0098] Right now

[0099] When M approaches positive infinity, substitute the feedback control u and the actual tracking displacement into formula (8) to obtain the limit That is, the optimal performance index of the piezoelectric ceramic driver model is obtained.

[0100] The comparison between the method described in this embodiment and the existing method is as follows:

[0101] In the past, the RK4 algorithm with a fixed step size required calculating k four times. i Function, the local error is O(h 5 ); while the fixed-step fifth-order Runge-Kutta (RK5) algorithm needs to calculate 10 times k i Function, the local error is O(h 6 ), although RK5 improves the calculation accuracy compared to RK4, it brings about the consumption of calculation cost. In order to integrate the above advantages and solve the related disadvantages, the method described in this embodiment combines the advantages of the fourth-order Runge Kutta method with the fifth-order Runge Kutta method, and proposes the RKF45 algorithm with adaptive step size, which only needs to calculate k six times ifunction, and adjust the error tolerance ε according to the need of accuracy, so that it is better than the RK4 algorithm with a fixed step size in terms of calculation accuracy and better than the RK5 algorithm in terms of calculation speed, thereby further improving the accuracy of piezoelectric actuator displacement tracking.

[0102] In order to prove the advantages of the method described in this embodiment, the system was simulated using the mathematical software MATLAB and experimental data were collected. The simulation time was set to 0.05 seconds. Figure 4 The error comparison between the present invention and the RK4 method when applied to the implementation example is shown in FIG. Figure 5 It is the displacement tracking error when the present invention is applied to the implementation example. It can be seen from the figure that the present invention significantly improves the displacement tracking accuracy of the system when solving the displacement tracking of the ECNLP model, and effectively optimizes the dynamic performance of the system.

[0103] The embodiments described above are only preferred embodiments of the present invention and are not intended to limit the scope of implementation of the present invention. Therefore, all changes made according to the shape and principle of the present invention should be included in the protection scope of the present invention.

Claims

1. A piezoelectric drive optimal trajectory tracking control method based on adaptive step size, characterized in that: include: The piezoelectric ceramic actuator is mathematically modeled using an electromechanical coupling nonlinear dynamic system model to obtain a piezoelectric ceramic actuator model; Convert the piezoelectric ceramic actuator model into a linear controllable canonical subsystem; The expected displacement x at time t = 0 * (0) as the initial actual tracking displacement x(0); The feedback control u is obtained through the linear controllable canonical subsystem; Substitute the initial actual tracking displacement x(0) into the linear controllable gauge subsystem and use the Runge-Kutta-Feldberg method with adaptive step size to calculate the actual tracking displacement; The performance index function of the linear controllable specification subsystem is determined, and the feedback control u and the actual tracking displacement are substituted into the performance index function to obtain the optimal performance index of the piezoelectric ceramic actuator model.

2. The piezoelectric drive optimal trajectory tracking control method based on adaptive step size according to claim 1, characterized in that: After mathematically modeling the piezoelectric ceramic actuator using the electromechanical coupling nonlinear dynamic system model, the obtained piezoelectric ceramic actuator model is as follows: in, is the derivative of X, is the state variable, U in is the control variable, F is the state variable X and the control variable U of the piezoelectric ceramic actuator model in The relevant function matrix, G is the weight matrix, x is the actual tracking displacement, is the derivative of x.

3. The piezoelectric drive optimal trajectory tracking control method based on adaptive step size according to claim 2 is characterized in that: The transformed linear controllable canonical subsystem is as follows: in, u is feedback control.

4. The piezoelectric drive optimal trajectory tracking control method based on adaptive step size according to claim 3 is characterized in that: The formula for calculating feedback control u is as follows: u=-R -1 B T P LQ Z+R -1 B T G g (3) R is the weight matrix, constant matrix P LQ It is the only symmetric positive definite solution to the Riccati algebraic equation; A T P LQ +P LQ A-P LQ BR -1 B T P LQ +C T QC=0 (4) C is the piezoelectric ceramic driver parameter, and its value is 1.2×10 -6 F, matrix G g for: G g =[P LQ BR -1 B T -A T ] -1 C T Qx * (5) x * is the expected displacement.

5. The piezoelectric drive optimal trajectory tracking control method based on adaptive step size according to claim 4 is characterized in that: The actual tracking displacement is obtained by using the Runge-Kutta-Feldberg method with adaptive step size, including: A1. Set initialization parameters: the number of allowed iterations temp, the minimum scaling factor min_scale, the maximum scaling factor max_scale, the error tolerance ε, the step size h, the system initial state value x0 = x(0), the independent variable t = t0, i = 1; A2. Obtain the current state value and the generated error; A3, determine whether the error generated in the previous step A2 is equal to 0, if so, directly go to step A5, otherwise go to step A4; A4: Determine whether i is equal to temp. If so, proceed to step A5. Otherwise, Update the scaling factor s, step size h, time t, and i: in, If min_scale<s<max_scale, then err i is the current error, h = h·s, t = t+h, i = i+1; After updating, return to step A2; A5. Use the latest state value as the actual tracking displacement x(t).

6. The piezoelectric drive optimal trajectory tracking control method based on adaptive step size according to claim 5, characterized in that: The calculation process of the state value and the error corresponding to the i-th iteration include: k1=hf(t i ,x i ) Where h is the step size.

7. The piezoelectric drive optimal trajectory tracking control method based on adaptive step size according to claim 1, characterized in that: The performance index function of the determined linear controllable canonical subsystem is as follows: Among them, x * is the expected displacement, R and Q are weight matrices, u is the feedback control, and x is the actual tracking displacement.

8. The piezoelectric drive optimal trajectory tracking control method based on adaptive step size according to claim 7, characterized in that: Find the optimal performance indicators of the piezoelectric ceramic actuator model, including: Transform the performance indicator function into: Let M tend to positive infinity, according to Right now When M approaches positive infinity, substitute the feedback control u and the actual tracking displacement into formula (8) to obtain the limit That is, the optimal performance index of the piezoelectric ceramic driver model is obtained.

Images